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Talk:Iota function
This function may not be well-defined but it is an interesting thought experiment and wholly relevant to googology. FB100Z • talk • 18:12, June 5, 2013 (UTC) This function indeed is not well-defined. According to definition from article iota function set contains all functions \(f_a(n)=n+a\). Then \(I(n)>F_a(n)\) for all a, and we define \(F_a(n)=f_a(n)\). So Iota is undefined even for 1. But if we alter the definition so that we can only use functions definable with at most n English symbols and we can compose at most n functions, it all becomes finite. LittlePeng9 (talk) 18:56, June 5, 2013 (UTC) :In fact, defining f(n) as the largest function definable in n English symbols can lead to the paradox, called Barry's paradox. For example, given f(70), and I can write the definition: "the largest function definable in 1000000000 English symbols", and it took 41 symbols. We get that f(70) >= f(1000000000), but the same applies for f(100000000), it can be shown that it actually > than \(f(10^{10^7})\), etc. The value of f(70) shall be undefined. Probably we can avoid this if we restrict f(n) as the largest function well-definable in n English symbols, it can work and get to order type \(\omega_1\). Ikosarakt1 (talk ^ ) 08:21, June 6, 2013 (UTC) :I thought about it and here is something that might help: every English sequence which doesn't properly define natural number is evaluated to 0. I guess it still leads to paradox, but this is property of language we use. LittlePeng9 (talk) 19:56, June 6, 2013 (UTC) I believe that I(n) is an attempt to create a function with order type \(\omega_1\). Ikosarakt1 (talk ^ ) 19:00, June 5, 2013 (UTC) :Well, given that every countable ordinal can be defined in English (which may follow from its inconsistency) whole fast-growing hierarchy can be defined in English, so Iota ( if well defined) would be all above this. But, as Deedlit once pointed out, existence of \(f_{\omega_1}(n)\) is independent of ZFC, so Iota is well out of its reach. LittlePeng9 (talk) 19:16, June 5, 2013 (UTC) :In that case, maybe Hollom tried to reach absolute infinity (which doesn't exist)? Also, an example of similar function is f(n), which is defined as the largest number defined in year n. Ikosarakt1 (talk ^ ) 19:31, June 5, 2013 (UTC) :Not exactly, if we had \(f_{\omega_1}(n)\) we could continue FGH for numbers with countable cofinality. But even then absolute infinity is far from our goal, as there is only continuum of N->N functions, and continuum is not absolute infinity. The exact answer of where FGH can't reach is connected to so called bounding number, for which most questions about size are independent of ZFC. And I'm sure your f(n) is eventually constant ;) LittlePeng9 (talk) 19:57, June 5, 2013 (UTC) :I'm very hope that f(n) shall be not a constant from some n. Ikosarakt1 (talk ^ ) 08:10, June 6, 2013 (UTC) :According to most fate-of-the-Universe theories either at some point Universe will collapse or entropy will reach its limits, so no more work will be done. LittlePeng9 (talk) 19:56, June 6, 2013 (UTC) I have a simpler definition. Instead of composing all the functions in the iota set, why not just pick the one that outputs the largest number? FB100Z • talk • 21:09, June 5, 2013 (UTC) :What means "composing" functions? If given \(f_1(n)\) and \(f_2(n)\), there are two ways to compose them: \(f_1(f_2(n))\) and \(f_2(f_1(n))\). Ikosarakt1 (talk ^ ) 08:13, June 6, 2013 (UTC) ::This is why we have choice of order in which we compose functions. LittlePeng9 (talk) 10:28, June 6, 2013 (UTC) I think I've fixed the definition to make it better defined. I have specified that it can do nothing that would lead to a circular reference and therefore infinity. It is the biggest number producable by combining every explicitly defined integer function ever (not every function must be used). The key is the word explicit. For example, until I mentioned it here, it is very unlikely that f_736467944635(n) in the fgh would have been used, because no one would have explicitly mentioned it before.DrCeasium (talk) 19:18, June 6, 2013 (UTC) : But my point on why function doesn't evaluate for any n still applies... I guess there isn't much improvement possible though. LittlePeng9 (talk) 19:56, June 6, 2013 (UTC) ::There shouldn't be a paradox because I specifically stated in the definition that it cannot do anything that would lead to a circular reference. The example I gave on the website, that there were only three functions in existence, f(x), g(x) and h(x), the value of I(x) was h(f(g(x))), and due to the rule about circular references, this will not change now that the iota function is also produced. The only difference between this example and the real world is that in real life there are a lot more functions.DrCeasium (talk) 16:20, June 7, 2013 (UTC) ::Ah, now I understand your point; Iota function set concerns defined functions, not all definable ones. It makes sense now. LittlePeng9 (talk) 17:44, June 7, 2013 (UTC) ::Also, defined by who? If the Universe is infinite, or there are infinite number of Universes in the Multiverse, there will be infinitely many googologists, and infinitely many functions defined by them. Ikosarakt1 (talk ^ ) 19:40, June 14, 2013 (UTC) :::I think we can limit ourselves to observable universe only, at least for now. LittlePeng9 (talk) 19:51, June 14, 2013 (UTC) I(n) isn't a constant Value of I(n) will change as soon as some new function will be defined. For example, I define: Ra2(n) = Ra(Ra(Ra(Ra(...(Ra(n))...)))) and I changed all values of I(n) just now. Ikosarakt1 (talk ^ ) 16:43, June 14, 2013 (UTC) :Yup, that's pretty much how this "function" works :P :Like lynz or ¥, the iota function is really a two-argument function; the other argument is time. For example, the iota function was different two days ago; we could call the earlier one \(I_{6/12/2013}(n)\) and the current one \(I_{6/14/2013}(n)\). Now the values of the function are constant, and iota is just within reach of a valid, formally defined function. :I propose a very strange (and almost formal) alternative to the iota function: the kappa function. It is defined by a deterministic cellular automaton that simulates a universe, allowing life to evolve and eventually creating virtual googologists. The kappa function \(\kappa(t)\) is defined as the largest number invented by these organisms at time \(t\). Obviously \(\kappa(t)\) won't even be defined until we get to some very large values of \(t\); but after it gets going I suspect its growth rate is pretty close to iota. FB100Z • talk • 17:11, June 14, 2013 (UTC) :I just thought how weird it'd be if we ever simulate such automaton for time suffiscently long that "people" living there would solve some problems in mathematics (or anything else) before we do so. Solving Riemann hypothesis by watching how cells interacting with their neighbours only either live or die. LittlePeng9 (talk) 18:33, June 14, 2013 (UTC) ::Yeah. Little dots moving around become living beings with their own minds, emotions, and relationships — the very idea of running such a simulation kind of freaks me out. But can a collection of theoretical dots have a sense of self, of consciousness? What is self, even? METAPHYSICS AAAAAAAAAAAA FB100Z • talk • 19:59, June 14, 2013 (UTC) ::I do, however, like the idea of having a virtual civilization solve our unsolved problems for us (or at least offer different perspectives). I do not like the idea that we might be such a simulation. ::This topic also reminded me of an Onion article. FB100Z • talk • 20:04, June 14, 2013 (UTC) :::so using all of these ideas about how ridiculously large the iota function can be, this is an interesting idea: the limit of the 2 input iota function, as mentioned above, including the possibility of going into the future is the most powerful function that ever will, has or can be, because it includes every function ever defined by anything in the observable universe at any point in time including the future.. Therefore, the function I_{\omega}(n) is completely unbeatable without just adding one or doing something else along those lines. DrCeasium (talk) 08:31, June 15, 2013 (UTC) :::BUT, from the point I mentioned somewhere else, this function will probably become eventually constant (in terms of change of time) because of heat death of the Universe, or any other disease, which will cause all googologists to be unable to produce new functions. I know a bit of time will pass until that, but it might be true. LittlePeng9 (talk) 08:51, June 15, 2013 (UTC) Order type So, is order type of I(n) is larger than, say \(\omega_2, \omega_\omega, \omega_{\omega_\omega}\), etc...? If not, then how large that ordinal can be? Ikosarakt1 (talk ^ ) 18:28, June 14, 2013 (UTC) :Speculation: The order type would be the limit of human thinking. Some large, countable ordinal that is simply too large for us to even define. Assuming this theory, this means that googology really is finite and there's a limit to how powerful we can make our functions. :My personal worldview is that human thinking and reason are fundamentally limited; I think that science and human reason can't explain the entire universe, but it's the most powerful and useful system we have. FB100Z • talk • 19:53, June 14, 2013 (UTC) My worst nightmare - definability theory meets ordinal analysis. Because there are only countably many definitions and MUCH more ordinals, there exists set (proper class, actually) of undefinable ordinals. From well ordering there must be unique smallest undefinable ordinal... which we just defined. I guess this contradictory ordinal is what we can call "limit of our thinking". LittlePeng9 (talk) 21:25, June 14, 2013 (UTC) :To avoid contradictions, you need to specify that definitions can't refer to the notion of definability in any way. Then we can talk about the smallest countable ordinal not definable with the restriction specified above, and it's not a contradiction. :We can refer to the above type of definitions as type-0 definitions, then we can define type-1 definitions as definitions that can refer to type-0 definitions, but not type-1 definitions. We can then define type-alpha definitions for any ordinal alpha, and define D_alpha as the smallest ordinal not definable with type-alpha definitions, without contradiction. Then here is a large ordinal: the smallest ordinal alpha such that alpha = D_alpha. Deedlit11 (talk) 23:25, June 14, 2013 (UTC) New, finalized function I have been thinking about how to make this function as powerful as possible, and I have come up with a pretty good idea. It is pretty mind blowing. The iota function needs to take 2 variables so that it is constant: one which is the time (in planck times) after the first instant of the year 1CE (or AD)(there was no year 0)(that is at the crossover point between 1BCE and 1CE)(the value could also be negative for times before this). I chose this time because if we were to use the beginning of the universe, we do not know this to anywhere near the necessary level of precision to relate the number of planck times to a certain day. The iota function is therefore updated every planck time with every single-input function that humans have access to at that point, and also all previous iota functions (this could also in the future include functions defined by aliens that we have access to). This therefore will include any spoken functions, because they would be accessible in someone's brain at some point, so would be in a certain iota function, and so in every following iota function. Also included in the function set are all the multi-input functions. However, the iota function requires single input functions, so, for example if the function was {a,b,c,d} in BEAF, then for the iota function, the function f(n) = {n,n,n,n} would be used. These functions are then arranged around a central value, which is the second input of the iota function. They do not all have to be used, cannot be used more than once (unless the same function has been described twice), and are arranged in the way that produces the largest final output possible. Due to the eventual end of the universe, where it ceases to have any matter in at all, and therefore no time, the iota function stops increasing due to there being no more planck lengths. The value of the iota function at this point will be the fastest-growing function ever defined or observed by the human race throughout its entire history. Sorry if this is a bit long. DrCeasium (talk) 10:06, June 15, 2013 (UTC) I hope, due to the fast tech-progress, mankind can eventually invent even the way to prevent end of Universe. Ikosarakt1 (talk ^ ) 10:11, June 15, 2013 (UTC) 2nd Iota function I can also create a second Iota function, witch include the normal Iota function. Continue with a third, forth, fifth, sixth etc. Wythagoras (talk) 11:02, June 16, 2013 (UTC) :due to the new rule about working through time as well and including all previous versions of the iota function, I can't see how you would get a second one without diagonalising over the entire existence of the universe (which I sort of did above), which would lead to the function not evaluating due to it using functions that aren't even defined yet. With a little more work (will come out soon) the final value of the iota function (due to death of universe, it will have a final value), which I shall call Hollom's number, will be the largest number that ever has, does or will ever exist (Even if you just use the iota function + 1 or similar!), without just adding one etc to Hollom's number. (Wow. Have fun picking the remaining bits of your head off the floor, because that is truly mind blowing!) This means that when the universe ends, Hollom's number will be the largest number to ever have existed in human history (excluding naïve extensions)! DrCeasium (talk) 12:42, June 16, 2013 (UTC) :Universe is eternal. Ikosarakt1 (talk ^ ) 18:21, June 16, 2013 (UTC) :: scenario says something opposite. LittlePeng9 (talk) 18:32, June 16, 2013 (UTC) ::Doesn't it makes you depressed? Ikosarakt1 (talk ^ ) 18:34, June 16, 2013 (UTC) ::To be honest, I don't believe in Big Crunch scenario. Heat death is more realistic for me, and there is no escape from it. Well, maybe parallel universes... LittlePeng9 (talk) 18:42, June 16, 2013 (UTC) ::Sbiis Saibian computed (see under 10^249) that we have around 26 decillion years to come up with our survive. So, there are large amount of time to invent the way to prevent our extinction. Ikosarakt1 (talk ^ ) 18:49, June 16, 2013 (UTC) :::any method of surviving a heat death would have to go against the second law of thermodynamics, that entropy always increases, which works a bit like a clock counting down the loss of order until the unstoppable demise of the universe. Any survival mechanism would have to compete against infinity, which is a tough opponent to say the least. Another possibility is the big rip: expansion accelerates (it is at the moment) to the point where everything is violently ripped to shreds. This scenario happens much sooner (I think its around 10^20 years), and is completely unsurvivable, even for only a few seconds longer than other stuff. Time travel back in time is completely impossible, and if it could be done it would destroy the universe due to feedback. As for parallel universes, our current understanding is that it would require passing through outside a universe, also completely impossible due to outside of universes consisting of absolute nothing: no space, no time, no physics. As for it being depressing, yes, it is. Just try not to think about it too much is my tactic. DrCeasium (talk) 19:06, June 16, 2013 (UTC) ::I don't want to say Saibian is wrong, but theory I know (Great Unification theory) states protons decay into antielectron, not directly into pure energy. But even after that long time, even if we had way to recreate matter from energy, thermodynamics say that when entropy will hit maximum, energy wouldn't be reusable. As far as these laws go, there is no prevention of this within our Universe. LittlePeng9 (talk) 19:11, June 16, 2013 (UTC) :Is Hollom's number odd or even? :P LittlePeng9 (talk) 15:36, June 16, 2013 (UTC) ::Currently, that depends on what rayo's function (as the fastest growing function currently except for the iota itself) does. DrCeasium (talk) 15:50, June 16, 2013 (UTC) :::Oh, yeah, I just noticed that we actually can't know many aspects of this ultimate number, because if we knew them, we could define function which would alter final outcome such that it doesn't have given property. LittlePeng9 (talk) 16:53, June 16, 2013 (UTC) I'm currently working on another part of the definition that deals with all named numbers, meaning that nothing can escape the iota function, and hollom's number will be completely unbeatable. DrCeasium (talk) 18:19, June 16, 2013 (UTC)